The index of an algebraic variety

The index of an algebraic variety

Abstract.

Let be the field of fractions of a Henselian discrete valuation ring . Let be a smooth proper geometrically connected scheme admitting a regular model . We show that the index of can be explicitly computed using data pertaining only to the special fiber of the model .

We give two proofs of this theorem, using two moving lemmas. One moving lemma pertains to horizontal -cycles on a regular projective scheme over the spectrum of a semi-local Dedekind domain, and the second moving lemma can be applied to -cycles on an -scheme which need not be regular.

The study of the local algebra needed to prove these moving lemmas led us to introduce an invariant of a singular local ring : the greatest common divisor of all the Hilbert-Samuel multiplicities , over all -primary ideals in . We relate this invariant to the index of the exceptional divisor in a resolution of the singularity of , and we give a new way of computing the index of a smooth subvariety of over any field , using the invariant of the local ring at the vertex of a cone over .

KEYWORDS. Index of a variety, separable index, moving lemma, -cycles, -cycles, rationally equivalent, Hilbert-Samuel multiplicity, resolution of singularities, cone.

MATHEMATICS SUBJECT CLASSIFICATION: 13H15, 14C25, 14D06, 14D10, 14G05, 14G20

D.L. was supported by NSF Grant 0902161.
\newremark

definition[thm]Definition \newremarkques[thm]Question \newremarkremark[thm]Remark \newremarkexample[thm]Example \newremarkemp[thm]

Let be a non-empty scheme of finite type over a field . Let denote the set of all degrees of closed points of . The index of is the greatest common divisor of the elements of . The index is also the smallest positive integer occurring as the degree of a -cycle on . When is integral, let denote the regular locus of , open in . We note in LABEL:pro.deltareg that is a birational invariant of .

Let now be the field of fractions of a discrete valuation ring with residue field . Let . Let be a proper flat morphism, with regular and irreducible. Let be the generic fiber of . Write the special fiber , viewed as a divisor on , as , where for each , is irreducible, of multiplicity in . Using the intersection of Cartier divisors with -cycles on the regular scheme , we easily find that divides (see LABEL:defX/S). Our theorem below strengthens this divisibility, and shows that when is Henselian, the index of the generic fiber can be computed using only data pertaining to the special fiber.

Theorem LABEL:dx-ds Keep the above assumptions on .

  1. Then divides .

  2. When is Henselian, then .

Theorem LABEL:dx-ds answers positively a question of Clark ([Cla], Conj. 16). This theorem is known already when is a finite field ([Bosch-Liu], 1.6, see also [C-S], 3.1), or when is algebraically closed (same proof as in [Bosch-Liu]), or when is a curve with semi-stable reduction ([Cla], Thm. 9).

We give two proofs of Theorem LABEL:dx-ds, using two different moving lemmas which may be of independent interest. The first proof uses the Moving Lemma 2.1 stated below. A slightly strengthened version is proved in the text. The definition and main properties of the notion of rational equivalence of cycles are recalled in section 1.

Theorem 2.1 Let be a semi-local Dedekind domain, and let . Let be flat and quasi-projective, with regular. Let be a -cycle on , closure in of a closed point of the generic fiber of . Let be a closed subset of such that for all , has codimension at least in . Then is rationally equivalent to a cycle on whose support does not meet .

Our second proof of Theorem LABEL:dx-ds uses the Moving Lemma LABEL:mvsimple below, which allows some moving of a multiple of a cycle on a scheme which need not be regular. Recall (2) that is an -scheme if every finite subset of is contained in an affine open subset of .

Theorem LABEL:mvsimple Let be a noetherian -scheme. Let be a closed subset of of positive codimension in . Let . Let be a -primary ideal of , with Hilbert-Samuel multiplicity . Then the cycle is rationally equivalent in to a cycle such that no irreducible cycle occurring in is contained in .

Theorem LABEL:mvsimple is a consequence of a local analysis of the noetherian local ring found in section LABEL:section-local, and in particular in Theorem LABEL:local-mv. Our investigation of the local algebra needed to prove Theorem LABEL:mvsimple led us to introduce the following local invariant in LABEL:gammaA. Let be any noetherian local ring. Let denote the set of all Hilbert-Samuel multiplicities , for all -primary ideals of . Define to be the greatest common divisor of the elements of . Theorem LABEL:mvsimple and the definition of show that:

Corollary LABEL:mv-degree. Let be a reduced scheme of finite type over a field and let be a closed point. Then divides .

This statement is slightly strengthened when is not equidimensional in LABEL:pro.deltareg2. Recall that the Hilbert-Samuel multiplicity is the smallest element in the set , and that it is a measure of the singularity of the ring : if the completion of is a domain (or, more generally, is unmixed [HIO], 6.8 and 6.9), then is regular if and only if . The invariant is also related to the singularity of the ring : our next theorem shows that is equal to the index of the exceptional divisor of a desingularization of .

Theorem LABEL:thm.na and LABEL:cor.compute-n. Let be an excellent noetherian equidimensional local ring of positive dimension. Let , with closed point . Let be a proper birational morphism such that is regular. Let . Then .

Note that in the above theorem, the set of degrees and the set of Hilbert-Samuel multiplicities need not be equal, and neither needs to contain the greatest common divisors of its elements. The proof that involves a third set of integers attached to , which is an ideal in , so that the greatest common divisor of the elements of belongs to (LABEL:def-na). The proof shows that and . The properties of the invariant are further studied in section LABEL:perspective.

As an application of Theorem LABEL:cor.compute-n, we obtain a new description of the index of a projective variety.

Theorem LABEL:newdescriptionIndex. Let be any field. Let be a regular closed integral subscheme of . Denote by a cone over in . Let denote the vertex of the cone. Then .

The variant of Hilbert’s Tenth Problem, which asks whether there exists an algorithm which decides given a geometrically irreducible variety , whether has a -rational point, is an open question to this date ([Poo], p. 348). So is the possibly weaker question of the existence of an algorithm which decides, given , whether (LABEL:algorithm). In view of Theorem LABEL:newdescriptionIndex, we may also ask whether there exists an algorithm which decides, given a -rational point on a scheme of finite type , whether .

In the last section of this article, we settle a question of Lang and Tate [LT], page 670, when the ground field is imperfect, and prove:

Theorem LABEL:sep.index. Let be a regular and generically smooth non-empty scheme of finite type over a field . Then the index is equal to the separable index .

We thank the referee for a meticulous reading of the article and for many useful comments.

1. Rational equivalence

We review below the basic notation needed to state our moving lemmas. Let be a noetherian scheme. Let denote the free abelian group on the set of closed integral subschemes of . An element of is called a cycle, and if is an integral closed subscheme of , we denote by the associated element in .

Let denote the sheaf of meromorphic functions on a noetherian scheme (see [K], top of page 204 or [Liubook], Definition 7.1.13). Let . Its associated principal Cartier divisor is denoted by and defines a cycle on :

where ranges through the points of codimension in (i.e., the points such that the closure has codimension in ; this latter condition is equivalent to the condition ). The function is defined, for a regular element of , to be the length of the -module .

A cycle is rationally equivalent to ([K2], §2), or rationally trivial, if there are finitely many integral closed subschemes and principal Cartier divisors on , such that . Two cycles and are rationally equivalent in if is rationally equivalent to . We denote by the quotient of by the subgroup of rationally trivial cycles.

{emp}

We will need the following facts. Given a ring and an -module , we denote by the length of . Let be a noetherian local ring of dimension . Let be its minimal prime ideals. Let be the total ring of fractions of (with for ) and denote by the associated order function. Then

  1. Let , and let denote the image of in . Using [BLR], Lemma 9.1/6, we obtain that

  2. If is reduced, then the canonical homomorphism

    is an isomorphism.

  3. Let be a subring such that is finite. Let be the maximal ideals of , and let . Then

    Indeed, our hypothesis implies that the -module has finite length, so for a regular element , . Conclude using [FulBook], A.1.3, and the isomorphism .

{remark}

Let be the irreducible components of , endowed with the reduced structure. Let denote their generic points. Let , and let be the meromorphic function restricted to . If for all , only involves codimension points of , then is rationally equivalent to on , since (use 1 (1)).

However, in general is not rationally equivalent to . Consider for instance the projective variety over a field , union of and intersecting transversally at a single point . Let be a coordinate function on with . Let be a rational function on which restricts to on and is equal to on . Then is the -cycle , since the point does not have codimension in . It is clear however that is not rationally trivial in . This shows that the implication (1) (3) in the proposition in [Ful79], §1.8, does not hold in general.

A proper morphism of schemes induces by push forward of cycles a group homomorphism . If is any closed integral subscheme of , then , with the convention that if the extension is not algebraic. It is known ([K2], [Th], and 1 below) that in general further assumptions are needed for a proper morphism to induce a group homomorphism . This is illustrated by our next example, also used later in LABEL:ex.univcat, LABEL:rem.non-uc, and LABEL:rem.non-uc2. (This example contradicts [FulBook], Example 20.1.3.)

{example}

We exhibit below a finite birational morphism of affine integral noetherian schemes with regular, and a closed point of codimension with rationally equivalent to on , but such that is not rationally equivalent to on . The key feature in this example is that maps the point of codimension in to a point of codimension in . It turns out that is not universally catenary. Our example is similar to that of [EGA], IV.5.6.11. The idea of the construction of a ring that is not universally catenary by gluing two closed points of distinct codimensions is due to Nagata (see [Mat], 14.E).

Let be any field. Let be the field of rational functions with countably many variables. Consider the polynomial ring in one variable and the discrete valuation ring . Let . Let be an irreducible polynomial of degree . Let be the closed point corresponding to and let be the closed point corresponding to . Then and . The residue field is a finite extension of of degree , and .

Choose a field isomorphism . Let be the scheme obtained by identifying and via (see [Ped], Teorema 1, [Sch], 3.4, or [Fer], 5.4):

By definition, is the pre-image of the field under the canonical surjective homomorphism . The ideal of is then a maximal ideal of , defining a closed point whose residue field is isomorphic to . The inclusion induces a morphism .

It is easy to see that is finitely generated over . Since , we can thus produce a finite system of generators for the -module . More precisely, we have . Therefore, is finite and, hence, is noetherian by Eakin-Nagata’s theorem. The ring has dimension 2 and, thus, is catenary. The induced morphism is an isomorphism. Indeed, for any special open subset (i.e., , we have . So any fraction with is equal to with , and .

Fix now . Let . Then . By construction, induces an isomorphism , so that . We claim that is not rationally trivial on . Indeed, let be a closed integral subscheme of containing as a point of codimension . As and are of dimension , we must have . Let be the schematic closure of in , and let be the restriction of . Then is a finite birational morphism of integral noetherian schemes of dimension . The point cannot belong to , since otherwise the prime ideal would properly contain the prime ideal of height corresponding to the generic point of . Hence, .

Now let be a principal Cartier divisor on . Then, using 1 (3), . Therefore, if is rationally equivalent to , then . It follows that is not rationally equivalent to when ; in fact, has order in the group . The same proof shows that has order in the group . {emp} For general noetherian schemes, Thorup introduced a notion of rational equivalence depending on a grading on , which turns the quotient of by this equivalence into a covariant functor for proper morphisms and a contravariant functor for flat equitranscendental morphisms ([Th], Proposition 6.5).

We briefly recall Thorup’s theory below. A grading on a non-empty scheme is a map such that if , then ([Th], 3.1). A grading is catenary if the above inequality is always an equality ([Th], 3.6). An example of a grading on is the canonical grading . This grading is catenary if and only if is catenary and every local ring is equidimensional1 ([Th], p. 266) at every point.

Let be an integral closed subscheme of with generic point , and let . Denote by the cycle where we discount all components such that . One defines the graded rational equivalence on using the subgroup generated by the cycles , for all closed integral subschemes of . If is catenary, then the graded rational equivalence is the same as the usual (ungraded) one ([Th], Note 6.6). Denote by the (graded) Chow group defined by the graded rational equivalence.

Let be a morphism essentially of finite type. Let be a grading on . Then induces a grading on defined in [Th] (3.4), by

If is proper, then induces a homomorphism ([Th], Proposition 6.5). If is universally catenary and equidimensional at every point, and , then is a catenary grading on ([Th], 3.11). It is also true that if is universally catenary and is a catenary grading, then is a catenary grading on ([Th], p. 266, second paragraph). {emp} In particular, assume that both and are schemes of finite type over a noetherian scheme which is universally catenary and equidimensional at every point, and is a proper morphism of -schemes. Let and be two cycles on (classically) rationally equivalent. Then and are (classically) rationally equivalent on .

In Example 1, endow with the canonical grading, and with the grading . Then is catenary but is not, because has virtual codimension . Computations show that , generated by the classes of and . The group is isomorphic to , generated by the class of . The group is isomorphic to , generated by the classes of and , with the former of order . {emp} Let be a separated integral noetherian regular scheme of dimension at most . Let denote its generic point. Endow with the catenary grading (which is also the usual topological grading). Let be a morphism of finite type, and endow with the grading . This grading is catenary (1).

Let and let be such that . Then is an -cycle on . If and is a closed point , then is a subscheme of dimension of the fiber . If , then is dominant and . In the latter case, if and only if is semi-local and is contained in . Otherwise, .

In particular, the irreducible -cycles on are of two types: the integral closed subschemes of of dimension such that meets at least one closed fiber, and the closed points of contained in (in which case must be semi-local). We say that a -cycle is horizontal if its support is quasi-finite over , and that it is vertical if its support is not dominant over .

2. Moving Lemma for -cycles on regular with semi-local

Let be a quasi-projective scheme of pure dimension a field . Let denote the non-smooth locus of . The classical Chow’s Moving Lemma [Rob] and its generalization ([Con], II.9, assuming algebraically closed) immediately imply the following statement:

{emp}

Let . Let be a -cycle on with . Assume that . Let be a closed subset of of codimension at least in . Then there exists an -cycle on , rationally equivalent to , and such that .

Our goal in this section is to prove a variant of this statement for a scheme over a semi-local Dedekind base . An application of such a relative moving lemma is given in Theorem LABEL:dx-ds.

{emp}

Let be a scheme. We say that is an -scheme, or simply that is , if every finite subset of is contained in an affine open subset of . In particular, an -scheme is separated. The following examples of -schemes are well-known:

(1) Any affine scheme is . Any quasi-projective scheme over an affine scheme is ([Liubook], Proposition 3.3.36). More generally, a scheme admitting an ample invertible sheaf is ([EGA], II.4.5.4).

(2) If is , then any closed subscheme of is clearly . The same holds for any open subset of . Indeed, let be a finite subset of , then is contained in an affine open subset of . Hence, with quasi-affine. By (1), is contained in an affine open subset of .

(3) More generally, if is and is a morphism of finite type admitting a relatively ample invertible sheaf, then is . Indeed, any finite subset of has finite image in , so we can suppose that is affine. Then admits an ample invertible sheaf [EGA], II.4.6.6, and we are reduced to the case (1).

(4) A noetherian separated scheme of dimension is ([RayFA], Prop. VIII.1). Suppose is an algebraically closed field, and that is a regular -scheme of finite type. Let be a separated scheme of finite type. Then any proper -morphism is projective ([K66], Cor. 2).

For the purpose of our next theorem, we will call a noetherian integral domain a Dedekind domain if it is integrally closed of dimension or . A version of this theorem where is not assumed to be semi-local is proved in [GLL2], 7.2.

Theorem 2.1.

Let be the spectrum of a semi-local Dedekind domain . Let be a separated morphism of finite type, with regular and . Let be a horizontal -cycle on with finite over . Let be a closed subset of such that for every , any irreducible component of that meets is not an irreducible component of . Then there exists a horizontal -cycle on with finite, rationally equivalent to , and such that .

In addition, since is semi-local, consists of finitely many points, and since is , there exists an affine open subset of which contains . Then, for any such open subset , the horizontal -cycle can be chosen to be contained in , and to be such that if is any separated morphism of finite type with an open embedding over , then and are closed and rationally equivalent on .

Proof.

It suffices to prove the theorem when is irreducible and . Choose an affine open subset of containing . Since is closed in , it is affine.

Proposition LABEL:normalization-L-global shows the existence of a finite birational morphism such that the composition is a local complete intersection morphism (l.c.i). Clearly, when is excellent, we can take to be the normalization morphism, in which case is even regular, and LABEL:normalization-L-global is not needed. Since is affine and is finite, there exists for some a closed immersion .

Let . We claim that it suffices to prove the theorem for the -cycle and the closed subset in the affine scheme . Indeed, let be a horizontal -cycle whose existence is asserted by the theorem in this case. In particular, . Let be any open immersion over . Consider the associated open immersion and the projection . By hypothesis, and are closed and rationally equivalent in . One easily checks that because is birational. It follows from 1 that is rationally equivalent to on . Moreover, .

The existence of with the required properties follows from Proposition 2.2 below. Indeed, first note that since is l.c.i., each local ring , , is an absolute complete intersection ring ([EGA], IV.19.3.2). It follows that the closed immersion is a regular immersion ([EGA], IV.19.3.2).

Let . We note that since and for each point in over , is not an irreducible component of . Let be a closed point, and let . Then , , and . Our assumption on implies that the irreducible components of passing through have dimension at most . We can thus apply 2.2 below to conclude the proof of 2.1. ∎

Proposition 2.2.

Let be any semi-local affine noetherian scheme. Let be a morphism of finite type with affine. Let be an integral closed subscheme of , of codimension , and finite over . Suppose that the closed immersion is regular. Let be a closed subset of such that for all closed points , the irreducible components of that intersect all have dimension at most . Then there exists a cycle on rationally equivalent to and such that:

  1. The support of is finite over and does not meet . Moreover, for any closed point , does not contain any irreducible component of .

  2. Suppose that is universally catenary. Let be any separated morphism of finite type and let be any -morphism. Then is rationally equivalent to on .

Footnotes

  1. Recall that a ring of finite Krull dimension is equidimensional if for every minimal prime ideal of . A point is equidimensional if is.
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